[FOM] The boundary of objective mathematics

[joeshipman@aol.com]

-----Original Message-----
>From: Paul Budnik <paul at mtnmath.com>
>I an dividing mathematics into absolute objective statements and
>relative ones that are sometimes treated as if they were absolute.
....
>The difference I want to make is between an an arbitrary path that is
>followed by a recursive process in a potentially infinite universe and 
a
>completed infinite set.  This  is a distinction that goes back at 
least
>to Aristotle. My position is perhaps close to constructivists, but I 
do
>not demand a constructive proof of a statement. I only demand a
>constructive proof that all the events that determine the statement 
are
>themselves determined by finite events.

The practical attitude many mathematicians seem to take is that 
statements in the first-order language of arithmetic OR statements of 
higher type which have arithmetical consequences are meaningful. For 
example, Schanuel's conjecture cannot be formulated in arithmetic but 
using it one can prove that (e^(e^n)) is never an integer when n is an 
integer, a statement which can be given an arithmetical formulation in 
terms of convergence of computations. The existence of a (countably 
additive) real-valued measure on all subsets of the continuum is a 
statement of even higher type which has useful arithmetical 
consequences such as Con(ZFC).

Many mathematicians would also declare as meaningful statements those 
which are set-theoretically absolute (have the same truth value in all 
transitive models of ZFC). For example, the Invariant Subspace 
Conjecture (all bounded linear operators on Hilbert space have 
nontrivial invariant subspaces) does not have any arithmetical 
consequences that I know of, but is considered to be one of the major 
open problems in mathematics. (Of course one can't prove that the 
Invariant Subspace conjecture has no arithmetical consequences without 
proving it consistent, which might be no easier than proving it 
outright.)

My own view is that any statement about sets of bounded rank is 
meaningful, and that statements like GCH which involve universal 
quantification for sets of arbitrary rank are vague. In between these 
two classes of statements are existential statements with no bound upon 
the rank -- statements like "a measurable cardinal exists". (In other 
words, the statement "no measurable cardinal exists" is vague while "a 
measurable cardinal exists" is less so; this asymmetry is not 
unreasonable and is analogous the asymmetry between the statements "the 
Riemann Hypothesis is not provable" and "the Riemann Hypothesis is 
provable" which have different epistemological statuses).

-- JS